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Theorem isoml 35259
Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isoml.b 𝐵 = (Base‘𝐾)
isoml.l = (le‘𝐾)
isoml.j = (join‘𝐾)
isoml.m = (meet‘𝐾)
isoml.o = (oc‘𝐾)
Assertion
Ref Expression
isoml (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦
Allowed substitution hints:   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem isoml
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6411 . . . 4 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isoml.b . . . 4 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2851 . . 3 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6411 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 isoml.l . . . . . . 7 = (le‘𝐾)
64, 5syl6eqr 2851 . . . . . 6 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 4854 . . . . 5 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
8 fveq2 6411 . . . . . . . 8 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 isoml.j . . . . . . . 8 = (join‘𝐾)
108, 9syl6eqr 2851 . . . . . . 7 (𝑘 = 𝐾 → (join‘𝑘) = )
11 eqidd 2800 . . . . . . 7 (𝑘 = 𝐾𝑥 = 𝑥)
12 fveq2 6411 . . . . . . . . 9 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13 isoml.m . . . . . . . . 9 = (meet‘𝐾)
1412, 13syl6eqr 2851 . . . . . . . 8 (𝑘 = 𝐾 → (meet‘𝑘) = )
15 eqidd 2800 . . . . . . . 8 (𝑘 = 𝐾𝑦 = 𝑦)
16 fveq2 6411 . . . . . . . . . 10 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17 isoml.o . . . . . . . . . 10 = (oc‘𝐾)
1816, 17syl6eqr 2851 . . . . . . . . 9 (𝑘 = 𝐾 → (oc‘𝑘) = )
1918fveq1d 6413 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( 𝑥))
2014, 15, 19oveq123d 6899 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ( 𝑥)))
2110, 11, 20oveq123d 6899 . . . . . 6 (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 (𝑦 ( 𝑥))))
2221eqeq2d 2809 . . . . 5 (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 (𝑦 ( 𝑥)))))
237, 22imbi12d 336 . . . 4 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
243, 23raleqbidv 3335 . . 3 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
253, 24raleqbidv 3335 . 2 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
26 df-oml 35200 . 2 OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
2725, 26elrab2 3560 1 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089   class class class wbr 4843  cfv 6101  (class class class)co 6878  Basecbs 16184  lecple 16274  occoc 16275  joincjn 17259  meetcmee 17260  OLcol 35195  OMLcoml 35196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-oml 35200
This theorem is referenced by:  isomliN  35260  omlol  35261  omllaw  35264
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